Inequality

[TheAnuja55]

If x/|x|<x which of the following must be true about x ?

A. x>1
B. x>-1
C. |x|<1
D. |x|=1
E. |x|^2>1

answer : B

[jkaustubh]

If x/|x|<x which of the following must be true about x ?

A. x>1
B. x>-1
C. |x|<1
D. |x|=1
E. |x|^2>1

answer : B

the answer is B.

Here is the explanation,

x/|x|<x

we have two cases

CASE I

x/x<x

or 1<x-------------A

CASE II

x/(-x)<x

or -1<x-------------B

from A and B

we get

x>-1

hence the answer

[GMATGuruNY]

If X/|X| < X Which of the following must be true about X ?

a) X>1

b) X>-1

c) |X| < 1

d) |X| = 1

e) |X|^2 > 1

x/|x| < x

x < x|x|

0 < x|x| - x

0 < x (|x| - 1)

The CRITICAL POINTS are -1, 0 and 1.
These are the only values where x(|x|-1) = 0.
To determine the ranges where x(|x|-1) > 0, test one value to the left and right of each critical point.

Plug x = -2 into x/|x| < x:
-2/ |-2| < -2
-1 < -2.
Doesn't work.
x < -1 is not a valid range.

Plug x = -1/2 into x/|x| < x:
-1/2/ |-1/2| < -1/2
-1 < -1/2.
This works.
-1<x<0 is a valid range.

Plug x = 1/2 into x/|x| < x:
(1/2)/ |1/2| < 1/2
1 < 1/2
Doesn't work.
0<x<1 is not a valid range.

Plug x = 2 into x/|x| < x:
2/ |2| < 2
1 < 2.
This works
x > 1 is a valid range.

Thus, the valid ranges are -1<x<0 and x>1.

Since it's possible that x=-1/2:
Eliminate A, since it doesn't have to be true that x>1.
Eliminate D, since it doesn't have to be true that |x|=1.
Eliminate E, since it doesn't have to be true that |x|² > 1.

Since it's possible that x=2:
Eliminate C, since it doesn't have to be true that |x| < 1.

The correct answer is B.

Since both -1<x<0 and x>1 are to the right of -1, it must be true that x > -1.

[sathishkumarjva9888]

Mitch,

As per Answer choice b) X>-1, X can even include 1/2, which doesnot satisfy the question. Though, B is the better available as it includes both -1/2 and 2, would GMAT include such controversial answer choices?

Regards,
Sathish

[TheAnuja55]

Mitch,

As per Answer choice b) X>-1, X can even include 1/2, which doesnot satisfy the question. Though, B is the better available as it includes both -1/2 and 2, would GMAT include such controversial answer choices?

Regards,
Sathish

Sathish,

Look at the explanation in this manner:

Observe the critical points first
The CRITICAL POINTS are -1, 0 and 1.

Now look for the ranges in which the values gonna work.

1)x<-1
Plug x = -2 into x/|x| < x:
-2/ |-2| < -2
-1 < -2.
Doesn't work.
x < -1 is not a valid range.

2)-1<x<0
Plug x = -1/2 into x/|x| < x:
-1/2/ |-1/2| < -1/2
-1 < -1/2.
This works.
-1<x<0 is a valid range.

3)0<x<1
Plug x = 1/2 into x/|x| < x:
(1/2)/ |1/2| < 1/2
1 < 1/2
Doesn't work.
0<x<1 is not a valid range.

4)x>1
Plug x = 2 into x/|x| < x:
2/ |2| < 2
1 < 2.
This works
x > 1 is a valid range.

Hence we got the two valid ranges: -1<x<0 and x>1.

Now observe, what we can eliminate using the above information:

Since it's possible that x=-1/2:
Eliminate A, since it doesn't have to be true that x>1.
Eliminate D, since it doesn't have to be true that |x|=1.
Eliminate E, since it doesn't have to be true that |x|² > 1.

Since it's possible that x=2:
Eliminate C, since it doesn't have to be true that |x| < 1.

So only we are left with answer choice B.

[sathishkumarjva9888]

Yes Anuja, i agree that we are left with answer choice B as it satisfies both -1<x<0 and x>1. Can you provide the source of this question.

But my point is just to clarify whether GMAT would include such controversial answers whose range satisfies as well as vioaltes the question.

[TheAnuja55]

Yes Anuja, i agree that we are left with answer choice B as it satisfies both -1<x<0 and x>1. Can you provide the source of this question.

But my point is just to clarify whether GMAT would include such controversial answers whose range satisfies as well as vioaltes the question.

Some or the other time definitely, just you need to find that key point, then you are sorted. Like in this question, finding the ranges, lead us to eliminate answer choices.

I found this question, in a question bank which I downloaded from gmatclub.